Exact sequences in the cohomology of a group extension

TL;DR

This paper clarifies the relationship between a seven-term exact sequence in group cohomology and the spectral sequence associated with a group extension, linking previous results and spectral sequence maps.

Contribution

It establishes a precise connection between the maps in a known exact sequence and those from the spectral sequence in group cohomology.

Findings

Maps in the exact sequence correspond to spectral sequence maps

Links previous work to spectral sequence theory

Clarifies the role of specific maps beyond inflation and restriction

Abstract

In [J. of Alg. 369: 70-95, 2012], the authors constructed a seven term exact sequence in the cohomology of a group extension G of a normal subgroup N by a quotient group Q with coefficients in a G-module M. However, they were unable to establish the precise link between the maps in that sequence and the corresponding maps arising from the spectral sequence associated to the group extension and the G-module M. In this paper, we show that there is a close connection between [J. of Alg. 369: 70-95, 2012] and our two earlier papers [J. of Alg. 72: 296-334, 1981] and [J. Reine Angew. Math. 321: 150-172, 1981]. In particular, we show that the results in the two papers just quoted entail that the maps of [J. of Alg. 369: 70-95, 2012] other than the obvious inflation and restriction maps do correspond to the corresponding ones arising from the spectral sequence.